Squared Frequency-Kt/V

Murakami et al. Renal Replacement Therapy (2019) 5:8 https://doi.org/10.1186/s41100-019-0198-7

RESEARCH Open Access Squared frequency-Kt/V: a new index of hemodialysis adequacy—correlation with solute concentrations by computer simulation Kaya Murakami1, Kenichi Kokubo1,2,3*, Minoru Hirose1,2, Kozue Kobayashi1,2 and Hirosuke Kobayashi1,2

Abstract Background: In patients undergoing home dialysis, conventionally, Kt/V has been regarded as an index of the removal efficiency per dialysis session. However, more recently, it has been considered that the hemodialysis product (HDP), rather than the Kt/V, is better associated with the clinical symptoms and outcomes in patients undergoing short daily dialysis or nocturnal dialysis. Nevertheless, the HDP lacks a theoretical background, and it does not take into consideration the dialyzer clearance or patient’s body size. The aims of the present study were to clarify the theoretical validity of HDP focusing on its association with solute concentration by computer simulation and to propose a new index of hemodialysis adequacy.

Methods: We used compartment models and calculated the time course of urea and β2 microglobulin (β2MG) concentrations to determine the peak concentrations and time-averaged concentrations at varying dialysis frequencies (n =2–7 sessions/week) and durations of dialysis per session (t = 1 to 8 h dialysis sessions).

Results: It was found that the peak concentrations of urea and β2MG were significantly correlated with the HDP. Based on this, we theoretically extracted the factor related to the peak concentration and defined the squared frequency-Kt/V (sf-Kt/V), as a new index for determining hemodialysis adequacy (sf-Kt/V=n2Kt/V; K, clearance; V,solutedistribution volume; n,frequency;t, dialysis time); this index was well correlated with the peak concentrations of urea and β2MG, even when the values of K and V were changed.

Conclusions: Since the sf-Kt/V is an index that reflects peak concentrations of urea and β2MG, which takes into account the dialysis frequency, session duration, dialyzer clearance, and the body weight of the patient, it will be a very useful tool for determining appropriate dialysis schedules and dialysis conditions for individual patients. Keywords: Kt/V, Hemodialysis product, Adequacy of dialysis, Home hemodialysis

Background quality of life [7, 8], and lower cost [2, 3, 5]. In home Home hemodialysis has the advantage that dialysis sched- hemodialysis, since the dialysis schedule can be flexibly ule (frequency and duration of dialysis) can be changed at determined, a reliable index useful for determining the ap- the patients’ own convenience. Many modalities of dialysis propriate home dialysis modality, dialysis schedule (fre- suitable for home hemodialysis have been proposed, in- quency and duration), and dialysis treatment conditions is cluding short daily dialysis [1], nocturnal dialysis [2–5], required. and daily nocturnal dialysis [6]. Home hemodialysis has The Kt/V is a well-known index of the removal effi- been reported to be linked to increased survival [1, 7, 8], ciency per dialysis session. To compare different dialysis better blood pressure control [2], reduced left ventricular modalities, the standard Kt/V was proposed as an index mass [4], improved mineral metabolism [4], enhanced that can be uniformly used to measure and explicitly compare dialysis doses [9–11]. The use of this index was recommended by the National Kidney Foundation Kidney * Correspondence: [email protected] 1Kitasato University Graduate School of Medical Sciences, Sagamihara, Japan Disease Outcomes Quality Initiative (KDOQI) guidelines 2Kitasato University School of Allied Health Sciences, Sagamihara, Japan [12] in 2006 and also by the most recent KDOQI in 2015 Full list of author information is available at the end of the article

© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 2 of 11

[13], to evaluate the optimal dialysis dose and schedule for currently available, it is difficult to select a dialyzer and frequent dialysis; however, it was still not widely used, operating condition suitable for short daily dialysis or probably because (1) its calculation process is quite nocturnal dialysis on the basis of the HDP. complex; (2) the calculation required values of single-pool The purpose of the present study was to explore a new Kt/V and equilibrated Kt/V to fit the original definition of index based on solute removal useful for ascertaining standard Kt/V, a mass generation per unit volume of body the adequacy of hemodialysis that would incorporate fluid normalized by the concentration; and (3) there is dialysis schedule (frequency and duration), the dialyzer little advantage to use standard Kt/V as an alternative to clearance, and patient’s body size. To this end, we first Kt/V Daugirdas which has already been widely used in sought to identify factors (such as the time-averaged daily clinical practices for three times a week schedule. concentrations/peak concentrations) that are strongly It was also reported that the hemodialysis product correlated with the HDP using a kinetic model. From (HDP), rather than the Kt/V, is better correlated with the our results, we developed an index that was also well clinical symptoms and outcomes in patients undergoing correlated with these factors and validated the index short daily dialysis and nocturnal dialysis, the modalities using a kinetic model. most often used for home hemodialysis [14]. HDP is an index of the adequacy of hemodialysis proposed by Methods Scribner and Oreopoulos [14] in 2002 as their opinion. Kinetic model for urea This index incorporates the dialysis frequency as an im- We used the single-pool model (Fig. 1a) and calculated portant variable: the time course of the concentrations of urea to determine the peak concentrations and time-averaged con- HDP ¼ n2 t centrations under various dialysis schedules (Table 1). The change of the blood urea level during dialysis was where n is the dialysis frequency [times/week], and t is calculated using the following equation: the dialysis time per session [h/session]. VdC ðtÞ HDP has been proposed simply based on the very B ¼ −K C ðtÞþG dt u B u positive results with more frequent dialysis reported by K C ð Þ−G Kut G De Palma et al. [15], Buoncristiani et al. [16], Bonomini u B 0 u − V u That is, CBðtÞ¼ K e þ K . et al. [17], Pierratos et al. [2, 3], and Lockridge et al. u u [18]. The HDP has been shown to be well correlated And the change of the blood urea level outside the with the patients’ symptoms [14], and fewer symptoms period of dialysis was calculated by the following equation: [19–28] in dialysis patients has been reported in the treatment at HDP > 70, although these are retrospective VdC ðtÞ B ¼ −K C ðtÞþG observational studies where HDP has not been used as a dt u B u treatment criterion. Use of the HDP as an index of the Gu adequacy of hemodialysis has major limitations: it is an That is, CBðtÞ¼ V t þ CBðt1Þ empirical index that lacks a physiological or theoretical where CB is the blood urea level, CB(t1) is the blood background and ignores the effects of the dialyzer clear- urea level at the end of dialysis, Ku is the urea clearance, ance and patient’s body size. Therefore, although many and Gu is the endogenous production rate of urea. Using types of dialyzers (low flux to super high flux) are these equations, assuming Ku = 0, and a steady state (we

Fig. 1 Schematic diagram of single-pool (one-compartment) model (a) and three-compartment model (b) Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 3 of 11

Table 1 Setting conditions in the simulation Group A4 A5 A6 A7 A8 B4 B5 B6 B7 B8 C4 C5 C6 C7 C8 D3 D4 Dialysis frequencies [sessions/week] 2 2 2 2 2 3 33334444455 Dialysis time per session [h/session] 4 5 6 7 8 4 56784567834 HDP 16 20 24 28 32 36 45 54 63 72 64 80 96 112 128 75 100 Group D5 D6 D7 D8 E2 E3 E4 E5 E6 E7 E8 F1 F2 F3 F4 F5 F6 Dialysis frequencies [sessions/week] 5 5 5 5 6 6 66666777777 Dialysis time per session [h/session] 5 6 7 8 2 3 45678123456 HDP 125 150 175 200 72 108 144 180 216 252 288 49 98 147 196 245 294 Time course of changes in the solute concentrations for various dialysis schedule was calculated (the dialysis frequency varied from 2 to 7 per week and/or dialysis time per session from 1 to 8 h) repeatedly calculated the concentrations until the con- Runge-Kutta routine, using reported values [30] centrations at the beginning and end of the week were (Table 3). The time courses of the solute concentrations equal), the time courses of the urea concentrations were were calculated by the treatment conditions, such as the calculated by the treatment conditions and the produc- mass transfer coefficient between pools, solute production rate of urea [29] (Table 2). tion, and clearance of dialyzer. We compared the correlation between the peak

Kinetic model for β2-microglobulin concentrations/time-averaged concentrations of solutes Three-compartment models (Fig. 1b) in which β2-micro- and the HDP, total dialysis time, or squared frequency-Kt/ globulin (β2MG) was distributed in a slowly moving pool V (sf-Kt/V), by determining the coefficients of determin- and fast moving pool were used to calculate the time ation (R2), likelihood (L), and Akaike Information courses of β2MG concentration under various dialysis Criterion (AIC). schedules (Table 1). The values of total clearance, intercompartmental clearance parameters, and compartmen- Model fitness 2 tal volumes reported by Odell et al [30] for β2MG were Regarding HDP (n t) and total dialysis time (nt), we com- used. The equation of mass transfer in each compart- pared which one better estimates peak concentration and ment could be expressed by the following equations: time-averaged concentration. Wilcoxon’s signed-rank sum test was used to check whether there was a significant dif- V dC 1 1 ¼ −ðÞK þ K þ K C þ K C þ K C ference in the absolute value of the difference between the dt 01 12 13 1 12 2 13 3 þ G approximate curves calculated by a least squares method β and the calculated values by simulation. V dC Likelihood ratio test was used to compare the HDP 2 2 ¼ K C −K C n2t sf-Kt V dt 12 1 12 2 ( , degree of freedom is 2) and / (degree of freedom is 4). We compared the fitness between the HDP V dC sf-Kt V 3 3 ¼ K C −K C or the / and the peak concentrations of small mol- dt 13 1 13 3 ecules at varying dialyzer clearances (Fig. 4a, b), and the fitness between the HDP or the sf-Kt/V and the K 01 ¼ K r þ K nr þ K d Table 3 Treatment conditions used to calculate the time courses V d ¼ V 1 þ V 2 þ V 3 of β2MG concentration using a three-compartment model K K K where r, nr, and d are the renal, non-renal, and Clearance of β2MG, Kβ 40, 60, 80 [mL/min] dialyzer clearance, respectively. The distribution volume, Non-renal clearance, Knr 2.82 [mL/min] Vd, is the sum of the compartmental volumes, and Gβ is Renal clearance, Kr 0 [mL/min] the endogenous production rate of β2MG. The Intercompartmental clearance, K 75 [mL/min] differential equations were solved numerically using a 12 Intercompartmental clearance, K13 28.8 [mL/min] G Table 2 Treatment conditions used to calculate the time Endogenous production rate, β 0.159 [mg/min] courses of urea concentration using a single-pool model Fluid volume before the start of dialysis, V 36,000 [mL]

Clearance of urea, Ku 90, 120, 150, 180 [mL/min] Volume of compartment 1 per body weight 53 [mL/kg]

Endogenous production rate of urea, Gu 6.2 [mg/min] Volume of compartment 2 per body weight 39 [mL/kg] Fluid volume before the start of dialysis, V 36,000 [mL] Volume of compartment 3 per body weight 109 [mL/kg] Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 4 of 11

time-averaged concentrations of urea and β2MG at vary- strongly correlated with the peak concentrations of urea ing dialyzer clearances (Fig. 4c, d). and β2MG. First, the maximum log likelihood (L) was calculated The “unphysiological nature” of dialysis may also be from the residual sum of squares of the approximate demonstrated by the difference between the maximum curves and the calculated values by simulation by the and minimum concentrations. This parameter was, how- following equation. ever, not well correlated with HDP (data not shown). We propose HDP × K/V (n2t × K/V)asanewindex sf-Kt V N ÀÁcalled / (Appendix). This index incorporates both logL ¼ − logðÞþ 2π logσ2 þ 1 solute removal (K) and fluid volume of the patient (V), 2 whereas the HDP does not. We then investigated whether where σ2 is the maximum likelihood estimator of vari- this index would still be well correlated with the peak con- ance, and N is sample size. The difference of the devi- centrations of small molecules when the clearance of the ance was calculated by multiplying log L by − 2, dialyzer was varied. The correlation of the sf-Kt/V with the concentrations showed higher R2 and lower AIC than that of HDP, and the difference in maximum log likeli- ΔD ¼ − L −ð− L Þ 2log HDP 2log sf−Kt=V hood (deviance) was statistically large, meaning that the sf-Kt/V showed significantly better correlations with the ΔD We performed a chi-square test for with a differ- peak concentrations of small molecules than HDP (Fig. 4). ence of 2 degrees of freedom to compare HDP with In other words, while the correlations with the HDP be- sf-Kt V / regarding which one would be better for fitting came weak when we used a different dialysis clearance, β to the peak concentrations of urea and 2MG at varying the sf-Kt/V still showed strong correlations with the peak dialyzer clearances. concentrations of urea. Since sf-Kt/V is obtained by multi- AIC was also calculated by the following equation: plying HDP by K/V, that is, the dialyzer clearance and patient’s body weight is also considered in the calculation of sf-Kt V AIC ¼ −2L þ 2 Â ðÞnumber of parameters the / ; this index is deemed as being a useful new index for comparing the adequacy of dialysis based on sol- When the AIC is smaller, the model is considered to ute removal (dialysis dose). be a better fitting model after taking into account the While no precise threshold value of the HDP has been number of parameters used. determined based on clinical evidence, HDP > 70 proposed by Scribner and Oreopoulos [14] is well accepted Results as being indicative of adequate dialysis. In a patient with Search for factors correlated with the HDP a dry weight of 60 kg (body fluid volume, 36 L) undergo- We assumed that if we could identify the factors that ing dialysis with a dialyzer providing an average urea were more strongly correlated with the HDP than with clearance of 160 mL/min and HDP > 70 will be equiva- the total weekly dialysis time, we would be able to esti- lent to sf-Kt/V > 18.7. mate why HDP (n2t) is better correlated with the clinical Assuming that HDP > 70 is the threshold value for symptoms than the total weekly dialysis time (nt). We determining the adequacy of dialysis, we would deter- focused on the peak concentrations and time-averaged mine the treatment schedules at varying dialyzer concentrations of solute molecules per week. We used a clearances (Fig. 5) and varying patient body weights single-pool model to investigate the kinetics of urea and (Fig. 6) that would yield sf-Kt/V values of over 20. a three-compartment model to investigate the kinetics of When conducting dialysis four times a week, 6-h dia- β2MG [30]. lysis sessions are enough to obtain an HDP value of The time courses of solute concentrations under vari- over 70 (42 × 6 = 96; > 70), regardless of the dialyzer ous dialysis schedules (varying dialysis frequencies from urea clearance. When sf-Kt/V is used, and if patient 2 to 7/ week and varying dialysis times per session from weight was 60 kg, it can be judged that sufficient sol- 1 to 8 h/session) were calculated (Table 1). We deter- ute removal would be obtained when the dialyzer mined the peak concentrations and time-averaged con- clearance is 180 mL/min or 150 mL/min, but equiva- centrations for each solute. lent solute removal cannot be obtained with 6-h ses- The peak concentrations of all molecules were found sions when the dialyzer urea clearance is 120 mL/min to be significantly better correlated with the HDP than or 90 mL/min (Fig. 5), which would occur when blood with the weekly total dialysis time (Fig. 2). While the flow rate or dialysate flow rate set to lower value. time-averaged concentrations of urea were more tightly The appropriate dialysis frequency and dialysis time correlated with weekly total dialysis time than with HDP per session would also be calculated depending on (Fig. 3). Thus, our results suggested that the HDP is the patient’s body weight (Fig. 6). Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 5 of 11

Fig. 2 Peak concentrations vs. HDP and total weekly dialysis time. Relationships between the HDP and peak concentrations of small molecules (a)

and β2MG (b) for various dialysis schedules and the relationships between the total weekly dialysis time and peak concentrations of urea (c) and β2MG (d). The peak concentrations were significantly better correlated with the HDP than the total weekly dialysis time (p =0.0034forurea,p =0.0171 for β2MG)

Discussion in the patients [30](Table4), the percent changes of the The results of simulations revealed strong correlations peak (Table 5) and time-averaged concentrations (Table 6) of the HDP with the peak concentrations of urea and were determined. Intercompartmental clearances and vol- β2MG. Considering these results and the fact that the umes of the compartments changed the concentration by a HDP was empirically related to the clinical symptoms of few percentage points, and the endogenous production rate patients, from viewpoint of solute removal, a large value of β2MG changed the concentration by 25%. However, the of HDP would mean smaller peak concentrations, which difference among the schedules was only a few percentage could explain the reduction of the clinical symptoms. points,whichwouldbeexpectedtohavehadnosignificant In the field of pharmacokinetics, the toxicity of the influence on the correlation between the HDP and the peak substances is known to be time-dependent and concentration or time-averaged concentration. concentration-dependent [31]. The experience that Ultrafiltration and intercompartmental fluid shift may greater the HDP led to lighter the symptoms in dialysis somehow have an influence, especially the concentra- patients [14] and the results we obtained that peak con- tions of intermediate-sized molecules like β2MG. We centration of urea and β2MG better correlated with checked the effects of ultrafiltration (data not shown). HDP than total dialysis time suggest that the toxicity The results were similar to those obtained without tak- due to accumulated substances in dialysis patients may ing into consideration ultrafiltration, for both urea and concentration-dependent. β2MG. In regard to intercompartmental fluid shift, we Since the results obtained here may be dependent on the can discuss the results of the effects of the changes in parameters listed in Table 3,wehaveillustratedhowthere- the volumes of the compartments and intercompart- sults would change when other values are selected for the mental clearances. The differences in the peak and relevant parameters. When the parameter changed by about time-averaged concentrations among the schedules ob- 20%, which was the standard deviation of the measured data tained by varying the values of these parameters were Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 6 of 11

Fig. 3 Time-averaged concentrations vs. HDP and total weekly dialysis time. Relationships between HDP and the time-averaged concentrations of

urea (a)andβ2MG (b) for various dialysis schedules and relationships between the total weekly dialysis time and the time-averaged concentrations of urea (c)andβ2MG (d). The time-averaged concentrations of small molecules were significantly better correlated with the total weekly dialysis time than the HDP (p = 0.0004 for urea, p = 0.6261 for β2MG) only a few percentage points, which would be expected exponential decrease in the concentration was not taken to have exerted no significant influence on the correla- into consideration), we may approximate the above as: tions between the HDP and peak concentration or T ≒nKt: time-averaged concentration. EKR w The average weekly clearance per week, that is, EKR, This approximation has been shown as a good ap- is expressed using time-averaged concentration (TAC) proximation for thrice-weekly therapy [34]. Therefore, and generation rate as [32, 33]: using this approximation: EKR ¼ ðÞG=TAC : TAC ¼ GTw=nKt ¼ð1=ntÞðGTw=KÞ: If dialysis is performed n times per week with the same This approximation can lead to a difference from the clearance, same dialysis time, and same intervals be- actual value. The difference is considered to be attribut- tween the dialysis treatments, the concentration changes able to the intervals between dialysis treatments being dif- during the n dialysis sessions per week would be equal ferent (therefore, the concentrations at the beginning of and the total removal amount per week can be calcu- each dialysis session are not the same), and the fact that lated as: G, K,andV cannot be regarded as constant [34]. This equation shows that the time-averaged concentra- EKRT w ¼ nKt; tion of urea is inversely proportional to the total dialysis where Tw is the total number of minutes in the week = time per week (nt). It is obvious from the theoretical point 168 h. In actual clinical situations, although the intervals of view that the time-averaged concentration would be between dialysis treatments are not the same and the well correlated with the total dialysis time. In the simula- − TAC is estimated from limited sampled data (the tion of the present study, we obtained TAC = 630 t 1.0 in Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 7 of 11

Fig. 4 Peak concentrations vs. HDP and sf-Kt/V. Relationships between the HDP (a)orthesf-Kt/V (c) and the peak concentrations of urea at for varying

dialyzer clearances and relationships between the HDP (b)orthesf-Kt/V (d) and the peak concentrations of β2MG at varying dialyzer clearances. When we entered different dialysis clearances, the sf-Kt/V showed significantly better correlations with the peak concentrations of both urea and β2MG than HDP (p < 0.01 for both cases)

Fig. 5 Dialysis schedules that would yield sf-Kt/V values of over 20 for varying dialyzer clearances at constant V (V = 36,000 mL). Each line shows the schedule that would yield an sf-Kt/V of 20. The area above this line represents the dialysis times/frequencies that yield sf-Kt/V values of over 20. As the dialyzer clearance increases, the dialysis time/frequency needed to obtain an sf-Kt/V value of over 20 decreases Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 8 of 11

Fig. 6 Dialysis schedules that would yield sf-Kt/V values of over 20 for varying body weights of the patient at constant K (K = 180 mL/min). Each line shows the schedule that would yield an sf-Kt/V value of 20. The area above this line represents the dialysis times/frequencies that yield sf-Kt/V values of over 20. As the body weight of the patient increases, the dialysis time/frequency needed to obtain an sf-Kt/V value of over 20 increases. Thus, the optimal dialysis schedule depends on the patient’s body weight

Fig. 2d, implying that the dataset (dialysis frequency and comparisons of the clinical efficacy using sf-Kt/V as an in- dialysis time for each session) was selected with little bias dicator, and we propose to analyze clinical data in the fu- and the influence of time and frequency of dialysis on the ture to establish a satisfactory threshold of sf-Kt/V.Asthis peak concentration and average concentration can be ad- threshold value can be determined for each solute, it is ne- equately investigated using this dataset. cessary to clarify these thresholds not only for urea but Therefore, in this study, a new index sf-Kt/V was de- also for β2MG. vised considering the factors influencing the peak con- When we calculate sf-Kt/V, single-pool Kt/V can be 2 centrations, sf-Kt/V = n Kt/V.Thesf-Kt/V was used for urea and equilibrated Kt/V for β2MG as Kt/V. established as an indicator showing strong correlations For simple calculation to determine the dialysis sched- with the peak concentrations. Since sf-Kt/V is an indica- ule, dialysis time, and dialyzer clearance, we can use K tor based on solute removal that takes the dialyzer clear- as the dialyzer clearance corrected by the blood and di- ance and patient’s body size into consideration, it can be alysate flow rate, t as the dialysis time, and V as the esti- used to propose a dialysis schedule appropriate for each mated body fluid volume. However, if we use this simple individual patient and to determine the appropriate dia- calculation, sf-Kt/V would be affected by the differences lyzer clearance, dialysis time, dialysis frequency, etc. between the estimated and actual body fluid volume and Scribner and Oreopoulos recommended HDP > 70 between the calculated clearance and actual clearance based on the 30-year excellent survival outcome in Tassin caused by a decrease in the actual blood flow rate or re- [19–23]. HDP > 70 is equivalent to sf-Kt/V >20whenthe circulation. Therefore, when conducting cohort studies dialyzer used assumed to have an average urea clearance or randomized controlled trials to determine the thresh- is 160 mL/min and the average dry weight of the patient is old value of sf-Kt/V, it is considered better to use 60 kg. Therefore, assuming that HDP > 70 is a threshold single-pool Kt/V for urea and equilibrated Kt/V for value for determining the adequacy of dialysis, it can be β2MG for the calculation of sf-Kt/V. considered that an sf-Kt/V value of about 20 for urea may There were several limitations of the present study. This be an approximate threshold of an indicator of the ad- study was not a clinical study. Whether it is indeed an ap- equacy of dialysis. Currently, there are no data on propriate index is yet to be confirmed by a cohort study

Table 4 Values of the parameters of three-compartment model applied for measured data of β2MG in patients [30]

V1 [mL/kg] Knr [mL/min/kg] V2 [mL/kg] K12 [mL/min/kg] V3 [mL/kg] K13 [mL/min/kg] Gβ [mg/hour/kg] Mean ± SD 53 ± 9 0.047 ± 0.010 39 ± 11 1.25 ± 0.25 109 ± 27 0.48 ± 0.09 0.159 ± 0.041 Ratio of SD [%] ± 17 ± 21 ± 28 ± 20 ± 25 ± 19 ± 26 Data were expressed per kg body weight Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 9 of 11

Table 5 The percent changes of the peak concentrations determined by the calculation using maximum and minimum value (mean ± SD) of each parameter shown in Table 3 Dialysis schedule Percent changes of peak concentrations [%]

nt V1 Knr V2 K12 V3 K13 Gβ Maximum value (mean + SD) 2 7 − 1.76 − 11.30 − 2.07 − 0.02 − 3.28 − 0.56 25.79 34− 1.69 − 10.61 − 2.09 − 0.06 − 2.80 − 0.69 25.77 57− 2.45 − 6.76 − 3.18 − 0.04 − 4.58 − 1.02 25.89 74− 3.02 − 5.63 − 2.99 − 0.19 − 1.34 − 1.97 25.78 Minimum value (Mean − SD) 2 7 1.81 13.93 2.18 0.03 4.12 0.74 − 25.79 3 4 2.04 13.08 2.32 0.09 3.72 0.83 − 25.65 5 7 2.98 7.67 3.66 0.06 6.27 1.45 − 25.77 7 4 3.49 6.35 3.63 0.29 2.47 2.49 − 25.79 or randomized controlled trial. Furthermore, sf-Kt/V >20 concentrations more tightly than HDP. It offers the was also calculated by assuming that HDP > 70 is the promise of being a very useful index for determining ap- threshold value for determining the adequacy of the propriate dialysis schedules and dialysis conditions for dialysis schedule; however, HDP > 70 has not yet been con- individual patients. firmed by clinical trials whether this threshold value is appropriate or not. When K = 180 mL/min, V = 36,000 mL, the Appendix dialysis duration that would yield sf-Kt/V values of over 20 Development of a new index and its validation was calculated as 8 h, meaning Kt/V = 2.4. Current dialysis Taking into account the results that the peak concentra- guidelines for hemodialysis prescription [35]recommend tions of small molecules were better correlated with the Kt/V = 1.4 or more as the target dose for thrice-weekly dialy- HDP than with the total dialysis time and that the HDP sis. How to ensure consistency with the current guideline is has been shown to be well correlated with the clinical an important issue while using this index. sf-Kt/V is also not symptoms [14], we considered the peak concentrations apreciseindex,likethestandardKt/V or EKR, and does not of small molecules as the candidate factors closely re- reflect the residual kidney function. However, it does reflect lated to the clinical symptoms. the peak concentrations of the accumulated solutes in dialy- Using a simple kinetic model ignoring changes of the sis patients. With further clinical studies, this index has the body weight associated with water removal and intake, potential to become a better indicator for determining the the concentrations of the solutes just before dialysis for adequacy of dialysis in individual patients. thrice-a-week dialysis were calculated using the following equations: Conclusions The sf-Kt/V is an index that takes into account the dialy- C1 ¼ C0 exp ðÞþ−Kt=V GT 1=V sis frequency, session duration and clearance, and the C ¼ C ðÞþ−Kt=V GT =V body weight of the patient and correlates with the peak 2 1 exp 2

Table 6 The percent changes of time-averaged concentrations determined by the calculation using maximum and minimum value (mean ± SD) of each parameter shown in Table 3 Dialysis schedule Percent changes of time-averaged concentration [%]

nt V1 Knr V2 K12 V3 K13 Gβ Maximum value (mean + SD) 2 7 − 2.10 − 8.94 − 2.27 − 0.07 − 0.40 − 1.88 25.79 34− 1.88 − 8.92 − 1.89 − 0.18 0.65 − 1.75 25.75 57− 2.40 − 4.98 − 2.79 − 0.12 − 0.33 − 2.30 25.82 74− 2.79 − 5.08 − 2.30 − 0.33 1.29 − 2.23 25.74 Minimum value (mean − SD) 2 7 2.25 10.75 2.57 0.11 1.15 2.45 − 25.79 3 4 2.36 10.85 2.40 0.30 − 0.56 1.96 − 25.76 5 7 2.99 5.56 3.33 0.19 1.13 3.05 − 25.77 7 4 3.09 5.66 3.00 0.46 − 1.48 2.52 − 25.82 Murakami et al. Renal Replacement Therapy (2019) 5:8 Page 10 of 11

Abbreviations C0 ¼ C2 exp ðÞþ−Kt=V GT 3=V AIC: Akaike Information Criterion; CB: Blood level [mg/mL]; Cpeak: Peak concentration per week [mg/mL]; EKR: Average weekly clearance per week where K is the dialysis clearance [mL/min]; t is the dialy- [mL/min]; Gu: Endogenous production rate of urea [mg/min]; T T T G β sis time per session [min]; 1, 2, and 3 are the inter- β: Endogenous production rate of 2-microglobulin [mg/min]; K K dialysis times; V is the body fluid volume [mL]; and G is HDP: Hemodialysis product; : Dialysis clearance [mL/min]; d: Dialyzer clearance [mL/min]; K : Non-renal clearance [mL/min]; K : Renal clearance the generation of solutes [mg/min]. nr r [mL/min]; Ku: Clearance of urea [mL/min]; L: Likelihood; n: Dialysis frequency 2 If T3 is the maximum dialysis interval, C0 becomes the [sessions/week]; N: Sample size; R : Coefficients of determination; sf-Kt/ V: Squared frequency-Kt/V; t: Dialysis time per session for HDP [h/session]; peak concentration per week, Cpeak: t: Dialysis time per session [min/session]; T , T , T : Inter-dialysis times [min]; 1 2 3 TAC: Time-averaged concentration [mg/mL]; Tw: Total number of minutes in G −Kt −2Kt V V V V T þ T exp− þ T exp a week (10,080 min) [min]; : Body fluid volume [mL]; 1, 2, 3: Volume of V 3 2 V 1 V V compartments 1, 2, and 3 per body weight [mL/kg]; d: The sum of the Cpeak ¼ 2 − Kt compartmental volumes [mL]; β2MG: β2-microglobulin; σ : Maximum − 3 1 exp V likelihood estimator of variance

Acknowledgements Similarly, in the case of four-times-a-week dialysis, the Not applicable. following equation can be obtained. Funding G −Kt − Kt − Kt Institutional research fund. T þ T − þ T exp 2 þ T − 3 V 4 3 exp V 2 V 1 exp V C ¼ peak − Kt Availability of data and materials − 4 1 exp V Not applicable.

From these equations, in the case of the condition that Authors’ contributions Kt V C KM provided the research design, carried out the simulation and data analysis, the / is greater than 1, the peak concentration, peak, and wrote the manuscript. KeKo provided the working hypothesis, participated will become higher when the maximum dialysis interval in the research design, and wrote the manuscript. MH and KoKo participated in becomes longer, or when the dialysis frequency (n) de- the research design and substantially contributed to the study concept. HK Kt V provided the working hypothesis, participated in the research design, and creases or the / becomes less. substantially contributed to the study concept. All authors read and approved Therefore, the peak concentration would be well cor- the final manuscript. related with: Ethics approval and consent to participate Kt Not applicable. n 1 ; V T max Consent for publication Not applicable. where Tmax is the maximum interval between dialysis sessions [h]. Competing interests Kt The authors declare that they have no competing interests. This can be arranged into a dimensionless form as nð V Þ T ð w Þ ’ T max Publisher sNote Springer Nature remains neutral with regard to jurisdictional claims in published This index should be well correlated with the Cpeak.If the intervals between dialysis sessions are equal, then: maps and institutional affiliations. Author details T 1 T ≒ w ; Kitasato University Graduate School of Medical Sciences, Sagamihara, Japan. max n 2Kitasato University School of Allied Health Sciences, Sagamihara, Japan. 3Department of Medical Engineering and Technology, Kitasato University School of Allied Health Sciences, 1-15-1 Kitasato, Minami-ku, Sagamihara, and the following equation can be obtained. Kanagawa 252-0373, Japan. Kt T Kt K n w ≒n2 ¼ n2t Received: 27 September 2018 Accepted: 8 January 2019 V T max V V

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